Multi-indexed power series in ${\mathbb{C}}^m$ and their absolute convergence

Fix $m\ge 1$ and read ${\mathbb{C}}^m$ through Complex $m$-space and its real coordinate dictionary. A multi-index is $\alpha \in {\mathbb{N}}^m$, with $\mathrm{\mid }\alpha \mathrm{\mid }={\sum }_{k<m}{\alpha }_k$ and $\alpha !={\prod }_{k<m}{\alpha }_k!$ as in $C^k$ maps and multi-index derivative notation in Euclidean space, every index running over $k<m$ from $0$. For $w\in {\mathbb{C}}^m$ the complex monomial is

$w^{\alpha }:={\prod }_{k<m}{w}_k^{{\alpha }_k},$

a finite product in the multiplicative commutative monoid of $\mathbb{C}$ (The product $g_0{g}_1\cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n=0$) equal to the identity, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$) of natural powers in $\mathbb{C}$ (Integer powers in the complex field); for the zero multi-index $w^0=1$.

Enumerating the index set. $\mathbb{N}$ is countable and Every finite power of an at most countable set is at most countable makes ${\mathbb{N}}^m$ at most countable; it is infinite, so there is a bijection $\sigma :\mathbb{N}\to {\mathbb{N}}^m$ (Injection, surjection, bijection, Finite, countably infinite, countable, uncountable).

Let $c:{\mathbb{N}}^m\to \mathbb{C}$ and $a,z\in {\mathbb{C}}^m$. The multi-indexed power series ${\sum }_{\alpha }{c}_{\alpha }(z-a{)}^{\alpha }$ converges absolutely at $z$ when the complex series ${\sum }_n{c}_{\sigma (n)}(z-a{)}^{\sigma (n)}$ converges absolutely (Complex series, absolute convergence, complex power series, and radius of convergence) for one bijection $\sigma$, equivalently for every one. The two conditions agree, and the sums agree, because for bijections $\sigma ,\tau$ the series along $\tau$ is a rearrangement of the series along $\sigma$: applying Every absolutely convergent complex series converges, and rearrangements preserve its sum to the nonnegative series of moduli transfers convergence, and applying it again to the series itself transfers the sum. That common value is written ${\sum }_{\alpha }{c}_{\alpha }(z-a{)}^{\alpha }$ and no other notion of unordered sum is introduced.

Box partial sums. For $N\in \mathbb{N}$ put $B_N:=\{\alpha \in {\mathbb{N}}^m:{\alpha }_k\le N\text{ for every }k<m\}$, a finite set, and let $S_N(z):={\sum }_{\alpha \in B_N}{c}_{\alpha }(z-a{)}^{\alpha }$ be the corresponding finite sum (A finite sum in a commutative monoid indexed by an arbitrary finite set). If the series converges absolutely at $z$ with sum $S$, then $S_N(z)\to S$. Given $\epsilon >0$, absolute convergence supplies $n_0$ with ${\sum }_{n\ge n_0}\mathrm{\mid }{c}_{\sigma (n)}(z-a{)}^{\sigma (n)}\mathrm{\mid }<\epsilon$ and $\mid {\sum }_{n<n_0}{c}_{\sigma (n)}(z-a{)}^{\sigma (n)}-S\mid \le \epsilon$; taking $N$ large enough that $B_N$ contains $\sigma (0),\dots ,\sigma (n_0-1)$, every index of $B_N$ outside that finite list is $\sigma (n)$ for some $n\ge n_0$, so $\mathrm{\mid }{S}_N(z)-S\mathrm{\mid }\le 2\epsilon$. The same argument bounds $\mathrm{\mid }{S}_N(z)\mathrm{\mid }$ and the tail of the series by the corresponding tails of the series of moduli.

The series converges absolutely and uniformly on a set $S\subseteq {\mathbb{C}}^m$ when there are reals $M_{\alpha }\ge 0$ with $\mathrm{\mid }{c}_{\alpha }(z-a{)}^{\alpha }\mathrm{\mid }\le M_{\alpha }$ for every $z\in S$ and every $\alpha$, and with ${\sum }_n{M}_{\sigma (n)}$ convergent. By Weierstrass M-test for complex-valued function series the partial sums along $\sigma$ then converge uniformly on $S$ (Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary) and the series converges absolutely at every point of $S$.

Why a bijection is fixed rather than an unordered sum defined. The library already has one theory of complex series and one rearrangement theorem, and the clause above uses exactly those. Introducing a separate notion of summation over ${\mathbb{N}}^m$ would create a second convergence notion that every later statement would have to be matched against; instead every multi-indexed sum below means the sum of the one-variable series along any enumeration, which the rearrangement theorem makes unambiguous.

Where the series live. The natural regions here are the polydiscs of Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$ rather than balls. If every $\mathrm{\mid }{z}_k-a_k\mathrm{\mid }$ is positive, absolute convergence at $z$ controls the series on the closed polydisc with that polyradius. If some coordinate is zero, the same coordinatewise domination holds on the corresponding degenerate product set, but that radius vector is not called a polyradius. This is exactly the shape the kernel expansion and the Cauchy estimates on this page produce.